TM modes in Rectangular Waveguides
Introduction : -
In this topic we are going to analyze the propagation of TM modes in a rectangular waveguide.
Explanation : -
The TMmn modes in a rectangular guide are characterized by Hz = O. In other words, the z component of an electric field E must exist in order to have energy transmission in the guide. Consequently, the Helmholtz equation for E in the rectangular coordinates is given by :
A solution of the Helmholtz equation is in the form of
which must be determined according to the given boundary conditions. The procedures for doing so are similar to those used in finding the TE-mode wave.
The boundary conditions on Ez require that the field vanishes at the waveguide walls, since the tangent component of the electric field Ez is zero on the conducting surface. This requirement is that for Ez= 0 at x = 0, a, then Bn= 0, and for Ez= 0 at y = 0, b, then Dn = 0. Thus the solution as shown in Eq. (2) reduces to
where m = 1, 2, 3, . . ., and n = 1, 2, 3, . . .
If either m = 0 or n = 0, the field intensities all vanish. So there is no TM01 or TM10 mode in a rectangular waveguide, which means that TE10 is the dominant mode in a rectangular waveguide for a > b. For Hz = 0, the field equations, after expanding
are given by
These equations can be solved simultaneously for Ex , Ey ,Hx, and Hy in terms of Ez .The resultant field equations for TM modes are